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Lecture #3 The Process of Simulating Dynamic Mass Balances

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Outline Basic simulation procedure Graphical representation Post-processing the solution Multi-scale representation

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BASIC PROCEDURE

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Dynamic Simulation 1.Formulate the mass balances: 2.Specify the numerical values of the parameters 3.Obtain the numerical solution 4.Analyze the results dt dx =Sv(x;k) k i =value; I.C. x io =x i (t=0) Run a software package xixi t vivi Graphically: t

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Obtaining Numerical Solutions: use anyone you like – we use mathematica

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Set up the equations Specify parameters Solve Specify graphical output Plot and export Typical Mathematica Workbook Step 1 Step 2 Step 3 Step 4 Results

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The Results of Simulation: a table of numbers 123 450 xixi t Can correlate variables, r 2 off-set in time on a particular time scale

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GRAPHICAL REPRESENTATION

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Example Graphical Representation of the Solution, x(t)

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Multi-time scale representation: segmenting the x-axis

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Dynamic Phase Portrait Can also plot fluxes v i (t) and pools p i (t) in the same way; this is a very useful representation of the results

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Characteristic “Signatures” in the Phase Portrait

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POST-PROCESSING THE SOLUTION

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Post-processing the Solution 1. Computing the fluxes stst vivi t vivi vjvj t=o t ∞ v(t)=v(x(t)) dynamic responsephase portrait

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Post-processing the Solution 2. Forming pools; two examples Compute “pools”: p(t)=Px(t) t p (t) Example #2 ATP ADP AMP ADP then plot x1x1 x2x2 K eq =1 Example #1 total mass

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Post-processing the Solution 3. Computing auto correlations n=3 first segment n=m entire range n=m off-set in time, =2 Example 1 Example 2 Example 3

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MULTI-SCALE REPRESENTATION

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Tiling Phase Portraits x1x1 x2x2 x2x2 x1x1 x3x3 x1x1 k slope=k r 2 ~1 A series of tiled phase portraits can be prepared, one for each time scale of interest x1x1 x2x2 x3x3 x 1 vs. x 2 same as x 2 vs. x 1 symmetric array use corresponding locations in array to convey different information graph statistics

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Representing Multiple Time-Scales: tiling phase portraits on separate time scales Time: 0 -> 3 sec Time: 3 -> 300 sec

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Example System:

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Example of Time-Scale Decomposition time x 10 =1 x 20 =x 30 =x 40 =0 x 3, x 4 do not move Fast x 4 does not move Intermediate Slow

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All TimesFast IntermediateSlow Tiled Phase Portraits: overall and on each time scale expanded scale x 3 ~ 0 perfect correlation

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Post-processing the Solution 1. Forming pools p 1 and p 2 are dis-equilibrium variables p 3 is a conservation variable

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Tiled Phase Portraits for Pools: L-shaped; dynamically independent (1/2,1) (0,0) (1,1)(1,1/2) no correlations

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Summary Network dynamics are described by dynamic mass balances dx/dt=Sv(x;k) that are formulated after applying a series of simplifying assumptions To simulate the dynamic mass balances we have to specify the numerical values of the kinetic constants (k), the initial conditions (x), and any fixed boundary fluxes. The equations with the initial conditions can be integrated numerically using a variety of available software packages. The solution is in a file that contains numerical values for the concentration variables at discrete time points. The solution is graphically displayed as concentrations over time, or in a phase portrait. The solution can be post-processed following its initial analysis to bring out special dynamic features of the network. We will describe such features in more detail in the following three chapters.

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